How exactly does the Sigma parameter affect the Velvet shader?
It rather seems to define the response curve towards the view angle.

The velvet shader makes a material look brighter the smaller the angle between the view vector and the surface tangent vector is, it's bright if you look on a surface in the tangent direction and dark if you look at it from it's surface normal direction. 

With sigma=0, you seem to have to look at a surface perfectly in tangential direction to receive light, while higher values spread it across a wider range of angles so that with sigma=1, the distribution goes all the way to a normal view angle and is relatively even.